I told you two weeks ago that I thought there was some kind of cosmic connection between quantum mechanics and the formula for the surface area of the sphere. I got half-way through my description of a hypothetical spin-11 system when I decided to take a little break. Now it's two weeks later and I'm still pretty convinced there's some kind of connection there, but I can't quite nail it down.
Here's the thing. Quantum mechanics tells us you can fully describe the spin state of a spin-11 system by specifiying exactly 23 numbers, corresponding to all possible z-axis spins between plus and minus eleven. Be careful: I'm talking about a system where we know for a fact that the total spin is eleven. We just don't know the z-component. More specifically, we don't know the distribution of the z-components. There's nothing in quantum mechanics that says the system has to be in exactly one of those z-spin states. It just says that the complete spin state is fully described by listing those twenty-three numbers. (Complex numbers, actually, but that's besides the point.)
Notice that this "total spin state" is something much more complicated that what we call the classical angular momentum. This state we've created certainly has an angular momentum in the classical sense, but there's nothing that says it has to equal 11 Planck units (the dimensions of Planck's constant are the same as angular momentum), and there's nothing to say that it has to point in the z direction. This quantum system still has a "classical" angular momentum; if we know the 23 spin parameters we can easily calculate the classical spin; and it can be anything from zero to eleven, and it can be oriented along any axis.
Now I'm going to do something which I suspect is experimentally impossible but I really don't know: I want to restrict myself to cases where the total classical spin is 11 units, or at least very nearly so. This is a tiny fraction of all possible spin-11 systems. And then I want to pick a random axis in 3-dimensional space and call it my z-axis. And then I want to list the 23 complex numbers describing the spin state. Actually, at this point I want to convert from amplitudes to probabilities, which means squaring out the complex quantities so I have a list of real numbers between zero and 1.
Now I want to do the same thing for different random choices of the z-axis. So I have a hundred different lists of 23 numbers, numbered from +11 to -11. And now I want to ask: what are the average values? Is the average of spin-7 higher than the average of spin-4...or all the all simply equal to 1/23?
Remember, they can be interpreted as probabilities, so they have to add up to 1.
And here at last is the connection. If...and only if....the surface area of the sphere is equal to exactly four times the cross-sectional area...then all the probabilities must equal 1/23. I think I'm correct in this, but I'm not about to argue the steps. Qualitiatively, I'm trying to say that all z-values of spin are equally likely; but it turns out its very difficult to say precisely what you mean by a statement like that. I don't want to make things more complicated than they have to be, but this is honestly the best I've been able to do so far. I think it's some kind of cosmic connection but I'm still not quite sure...